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Please put the following argument in standard logical form: If either the premises are inconsistent or the conclusion is a tautology, then the argument is valid. If an argument is invalid, then it has consistent premises. The premises are inconsistent only if the conclusion is not a tautology. An argument is invalid if the conclusion is not a tautology. If the premises are consistent, then the argument is invalid. Therefore, if an argument is valid, then it has inconsistent premises. Please evaluate this argument. Is it valid? Explain in terms of a full truth table. If the conclusion is logically equivalent to a premise, can the argument be invalid? (Explain in terms of a truth table that consists of only the last premise and the conclusion.) Is the argument sound? Explain why each premise and the conclusion is true or false.

Hints:

1. 1. You want to rewrite the argument in standard logical form. This means that you will want to provide a dictionary and symbolize the argument. But the way, this is only one argument. Don’t break it down into multiple arguments. Pay attention to the material in 2.12 of the textbook.
2. 2. To keep down the number of statement letters, rely on the judicious use of negation (~). You only need three letters. If you use ‘V’ for ‘an argument is valid’, use ‘~V’ for ‘an argument is invalid’.
3. 3. If you find that the big truth table (the one with all five premises and one conclusion) distracts from the main text of your paper, you may place it in an appendix at the end of your paper. When you provide the big truth table, include the initial setup. I need to see this, especially if you skip steps. The smaller truth table should go in the main text; you are using it to explain why the logical equivalence of the conclusion to a premise does or does not guarantee validity.
4. 4. The truth table will tell you whether the argument is valid or invalid. In this case, it will not tell you whether the argument is sound. It may tell you that a sound interpretation is possible; but it won’t tell you that your argument corresponds to this sound interpretation. Fair warning. 5. If a statement is contingent, the truth table will not tell you whether it is true or false. You will have to plug in values and see what the statement actually says. Suppose you have the premise ‘S→V’ where ‘S’ stands for ‘an argument is a syllogism’ and ‘V’ stands for ‘an argument is valid’. To explain why ‘S→V’ is false, you need to explain that just because an argument is a syllogism does not mean it is valid (an example should be provided). The fact that ‘S→V’ is contingent in the truth table is irrelevant.
5. 6. To explain why the first premise is true, you need to explain why inconsistent premises guarantee validity and why a tautology as a conclusion guarantees validity (you need to explain both).
6. 7. When you discuss why the premises are true or false, it is a good idea to treat conditional statements as saying ‘If , then ’. It is easier to explain why the statement ‘If V, then S’ is true or false than discussing a statement such as ‘V only if S’ or ‘S, if V’. The statement ‘V only if S’ does not mean that the only way we can have V is to have S.
7. 8. When you discuss why the premises are true or false, it is a good idea to give a paragraph to each premise. That way, you are less likely to skimp on your discussion.
8. 9. We are relying on this definition for consistent premises: the premises are consistent if and only if there is an interpretation in which they are all true. (We are not confining our discussion to an argument with just one premise.) By the way, categorical syllogisms have consistent premises. Even if we have an instance in which the statements are false, that syllogism has other instances (and interpretations) in which the premises are true. So, do not give examples where you claim that a statement such as ‘All cats are dogs’ is a contradiction. It’s not. Keep in mind that categorical syllogisms generally do not include tautologies or contradictions.
9. 10. Use examples when relevant.
10. 11. Remember, good writing counts.

Outline of how the paper should look

1. I. Introduction
   1. This will be basic. You can start out by saying something like this: I will analyze the following argument in terms of validity and soundness.
2. II. Argument
   1. A. Present a dictionary
   2. B. Put the argument in standard logical form (no English)
3. III. Validity
   1. A. First, a big truth table. This includes every statement. It will have 8 rows. You can attach this truth table at the back of your paper, if you wish.
   2. B. Then a little truth table. This will have 4 rows. You can should put this in the main text of your paper. It will consist of the last premise and the conclusion.
   3. C. Validity
      1. 1. Explain why this argument is valid or invalid, using your big truth table
      2. 2. Explain why this argument is valid or invalid, using the little truth table What do we mean when we say two statements are logically equivalent?
      3. What do we need for an argument to be invalid?
4. IV. Premises
   1. 1. Why is the first premise true or false?
   2. 2. Why is the second premise true or false?
   3. 3. Why is the third premise true or false?
   4. 4. Why is the fourth premise true or false?
   5. 5. Why is the fifth premise true or false?
   6. 6. Why is the conclusion true or false?
      1. If a statement is not a contradiction or a tautology, the truth table is irrelevant. To explain why contingent statements are true or false, you need to look at the content, not the truth table!
5. V. Conclusion

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